Download Differential Equations.

Chapter Seven.
Mathematics II
Differential Equations.
Differential Equations.
A differential equation is an equation that contains one or more derivatives of a
differentiable function. An equation with partial derivatives is called a Partial Differential
Equation. While, an equation with ordinary derivatives, which is derivatives of a function of
a single variable, is called an Ordinary Differential Equation.
The order of a differential equation is the order of the equation’s highest order
derivative. A differential equation is linear if it can be put in the form
The degree of a differential equation is the power (exponent) of the equation’s highest
order derivative.
If F(x) in the equation above equal to zero the differential equation is called
homogeneous, otherwise it is called nonhomogeneous differential equation.
Solution of First Order Differential Equations.
1) Separable Equations.
A first order differential equations is separable if it can be put in the form:
To solve these equations, integrate M with respect to x and N with respect to y to
obtain an equation that relates y and x .
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Chapter Seven.
Mathematics II
Differential Equations.
2
Chapter Seven.
Mathematics II
Differential Equations.
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Mathematics II
Chapter Seven.
Differential Equations.
2) Linear First Order Equations.
A differential equation that can be written in the form:
is called a Linear First Order Equation. Where P and Q are functions of x.
Steps for Solving a Linear First Order Equation.
1. Put it in standard form and identify the functions P and Q.
2. Find the integrating factor:
𝐼. 𝐹 = 𝑒 ∫ 𝑃(𝑥)𝑑𝑥
3. Find y using the following equation:
𝑦(𝑥 ) =
1
∫ 𝐼. 𝐹 𝑄 (𝑥 )𝑑𝑥
𝐼. 𝐹
Sol.
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Chapter Seven.
Mathematics II
Differential Equations.